(CNN) -- Nebraska lawmakers voted Friday to change a controversial safe-haven law by restricting the age under which a child can be dropped off at a hospital without the parents being prosecuted.

On a vote of 43-5, the state Senate -- the only chamber in Nebraska's Legislature -- approved final passage of the revision. The change scraps the current version of the safe-haven law -- which has no age limitation -- and instead says that no child older than 30 days can be dropped off.

Nebraska Gov. Dave Heineman signed the revised version of the law.

All 50 states have safe-haven laws, intended to cut down on infanticide and the number of infants abandoned in unsafe locations, according to the Web site of the National Safe Haven Alliance, a group in support of such laws. Only the District of Columbia lacks such a law, the alliance says.

But unlike other state's laws, which establish a time limit for the infants to be given up, Nebraska's law merely said "child," which could be interpreted as anyone under the age of 18.

Thirty-five children -- all but six of them older than 10 -- have been dropped off at Nebraska hospitals since the law took effect in September, according to the state Department of Health and Human Services. Five came from other states, with parents traveling to Nebraska from Michigan, Indiana, Iowa, Florida and Georgia.

The Senate met in an emergency session to change the law. Legislators had expressed concern that if the law were not changed, the state's child welfare system would be overwhelmed by older children dropped off by parents around the country who found themselves unable to provide proper care.

"The abandonment of these children -- and the harm it is causing them -- is an immediate concern," said Jen Rae Hein, spokeswoman for Heineman.

Social worker Courtney Anderson said some children beg their parents or guardians not to abandon them. "They may not really understand why they are being left at the hospital. But they know they are being left, and the parent or guardian might be fleeing," she said.

On Tuesday, a 15-year-old girl was left at a hospital in Hall County, in the central part of the state. Last week, a 14-year-old boy and his 17-year-old sister were dropped off at an Omaha hospital; the girl ran away from the hospital. Earlier in the week, a father flew in from Miami, Florida, to leave his teenage son at a hospital, officials also said.

"Please don't bring your teenager to Nebraska," Heineman told CNN. "Think of what you are saying. You are saying you no longer support them. You no longer love them."

State Sen. Tom White said lawmakers were caught off guard by the number of teenagers taken in under the law.

"What you've seen is an extraordinary cry for help from people all across the country," he said. "Nebraska can't afford to take care of all of them. Nebraska would like to be able to, but they know that we can't so we are going to have to change the law."

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"We didn't think [the law] would be used to the extent it [has been]," state Sen. Brad Ashford said. "We didn't anticipate children coming from other states."

Tysheema Brown drove from Georgia to leave her teenage son at an Omaha hospital.

"Do not judge me as a parent. I love my son and my son knows that," Brown said. "There is just no help. There hasn't been any help."

Nebraska has 6,600 children in state custody; that per-capita rate is one of the nation's highest, said Todd Landry, director of the Division of Children and Family Services for the Department of Health and Human Services,

"I think this has spurred some really healthy conversations about, how do parents get the help that they need when they are struggling with some of these parenting issues?" he said.

"And the message that we have been trying to get out is, 'Don't wait until it's a crisis. Reach out to your family and friends.' "